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The behavior of this function is very similar to the MATLAB linkage function." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A " }, { "__type": "InlineMath", "__tag": 4057, "value": "(n-1)" }, { "__type": "Text", "__tag": 4046, "value": " by 4 matrix " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z" }, { "__type": "Text", "__tag": 4046, "value": " is returned. At the " }, { "__type": "InlineMath", "__tag": 4057, "value": "i" }, { "__type": "Text", "__tag": 4046, "value": "-th iteration, clusters with indices " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 0]" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 1]" }, { "__type": "Text", "__tag": 4046, "value": " are combined to form cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "n + i" }, { "__type": "Text", "__tag": 4046, "value": ". A cluster with an index less than " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " corresponds to one of the " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " original observations. The distance between clusters " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 0]" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 1]" }, { "__type": "Text", "__tag": 4046, "value": " is given by " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 2]" }, { "__type": "Text", "__tag": 4046, "value": ". The fourth value " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z[i, 3]" }, { "__type": "Text", "__tag": 4046, "value": " represents the number of original observations in the newly formed cluster. Note that the dtype of " }, { "__type": "InlineCode", "__tag": 4051, "value": "Z" }, { "__type": "Text", "__tag": 4046, "value": " is " }, { "__type": "InlineCode", "__tag": 4051, "value": "float64" }, { "__type": "Text", "__tag": 4046, "value": " even though columns 0, 1 and 3 contain integer data and should be interpreted as such. This is for historical reasons." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The following linkage methods are used to compute the distance " }, { "__type": "InlineMath", "__tag": 4057, "value": "d(s, t)" }, { "__type": "Text", "__tag": 4046, "value": " between two clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": ". The algorithm begins with a forest of clusters that have yet to be used in the hierarchy being formed. When two clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " from this forest are combined into a single cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " are removed from the forest, and " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " is added to the forest. When only one cluster remains in the forest, the algorithm stops, and this cluster becomes the root." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A distance matrix is maintained at each iteration. The " }, { "__type": "InlineCode", "__tag": 4051, "value": "d[i,j]" }, { "__type": "Text", "__tag": 4046, "value": " entry corresponds to the distance between cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "i" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "j" }, { "__type": "Text", "__tag": 4046, "value": " in the original forest." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "At each iteration, the algorithm must update the distance matrix to reflect the distance of the newly formed cluster u with the remaining clusters in the forest." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Suppose there are " }, { "__type": "InlineMath", "__tag": 4057, "value": "|u|" }, { "__type": "Text", "__tag": 4046, "value": " original observations " }, { "__type": "InlineMath", "__tag": 4057, "value": "u[0], \\ldots, u[|u|-1]" }, { "__type": "Text", "__tag": 4046, "value": " in cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "|v|" }, { "__type": "Text", "__tag": 4046, "value": " original objects " }, { "__type": "InlineMath", "__tag": 4057, "value": "v[0], \\ldots, v[|v|-1]" }, { "__type": "Text", "__tag": 4046, "value": " in cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": ". Recall, " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " are combined to form cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": ". Let " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": " be any remaining cluster in the forest that is not " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The following are methods for calculating the distance between the newly formed cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " and each " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='single' assigns" } ] }, { "__type": "Math", "__tag": 4058, "value": "d(u,v) = \\min(dist(u[i],v[j]))" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for all points " }, { "__type": "InlineMath", "__tag": 4057, "value": "i" }, { "__type": "Text", "__tag": 4046, "value": " in cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "j" }, { "__type": "Text", "__tag": 4046, "value": " in cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": ". This is also known as the Nearest Point Algorithm." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='complete' assigns" } ] }, { "__type": "Math", "__tag": 4058, "value": "d(u, v) = \\max(dist(u[i],v[j]))" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for all points " }, { "__type": "InlineMath", "__tag": 4057, "value": "i" }, { "__type": "Text", "__tag": 4046, "value": " in cluster u and " }, { "__type": "InlineMath", "__tag": 4057, "value": "j" }, { "__type": "Text", "__tag": 4046, "value": " in cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": ". This is also known by the Farthest Point Algorithm or Voor Hees Algorithm." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='average' assigns" } ] }, { "__type": "Math", "__tag": 4058, "value": "d(u,v) = \\sum_{ij} \\frac{d(u[i], v[j])}\n {(|u|*|v|)}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for all points " }, { "__type": "InlineMath", "__tag": 4057, "value": "i" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "j" }, { "__type": "Text", "__tag": 4046, "value": " where " }, { "__type": "InlineMath", "__tag": 4057, "value": "|u|" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "|v|" }, { "__type": "Text", "__tag": 4046, "value": " are the cardinalities of clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": ", respectively. This is also called the UPGMA algorithm." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='weighted' assigns" } ] }, { "__type": "Math", "__tag": 4058, "value": "d(u,v) = (dist(s,v) + dist(t,v))/2" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where cluster u was formed with cluster s and t and v is a remaining cluster in the forest (also called WPGMA)." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='centroid' assigns" } ] }, { "__type": "Math", "__tag": 4058, "value": "dist(s,t) = ||c_s-c_t||_2" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "c_s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "c_t" }, { "__type": "Text", "__tag": 4046, "value": " are the centroids of clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": ", respectively. When two clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " are combined into a new cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": ", the new centroid is computed over all the original objects in clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": ". The distance then becomes the Euclidean distance between the centroid of " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " and the centroid of a remaining cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": " in the forest. This is also known as the UPGMC algorithm." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='median' assigns " }, { "__type": "InlineMath", "__tag": 4057, "value": "d(s,t)" }, { "__type": "Text", "__tag": 4046, "value": " like the " }, { "__type": "InlineCode", "__tag": 4051, "value": "centroid" }, { "__type": "Text", "__tag": 4046, "value": " method. When two clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": " are combined into a new cluster " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": ", the average of centroids s and t give the new centroid " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": ". This is also known as the WPGMC algorithm." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method='ward' uses the Ward variance minimization algorithm. The new entry " }, { "__type": "InlineMath", "__tag": 4057, "value": "d(u,v)" }, { "__type": "Text", "__tag": 4046, "value": " is computed as follows," } ] }, { "__type": "Math", "__tag": 4058, "value": "d(u,v) = \\sqrt{\\frac{|v|+|s|}\n {T}d(v,s)^2\n + \\frac{|v|+|t|}\n {T}d(v,t)^2\n - \\frac{|v|}\n {T}d(s,t)^2}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " is the newly joined cluster consisting of clusters " }, { "__type": "InlineMath", "__tag": 4057, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "t" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "v" }, { "__type": "Text", "__tag": 4046, "value": " is an unused cluster in the forest, " }, { "__type": "InlineMath", "__tag": 4057, "value": "T=|v|+|s|+|t|" }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "InlineMath", "__tag": 4057, "value": "|*|" }, { "__type": "Text", "__tag": 4046, "value": " is the cardinality of its argument. This is also known as the incremental algorithm." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Warning: When the minimum distance pair in the forest is chosen, there may be two or more pairs with the same minimum distance. This implementation may choose a different minimum than the MATLAB version." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "build-install/usr/lib/python3.14/site-packages/scipy/cluster/hierarchy/_hierarchy_impl.py", "item_line": 723, "item_type": "function", "aliases": [ "scipy.cluster.hierarchy.linkage" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "from scipy.cluster.hierarchy import dendrogram, linkage\nfrom matplotlib import pyplot as plt\nX = [[i] for i in [2, 8, 0, 4, 1, 9, 9, 0]]\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "Z = linkage(X, 'ward')\nfig = plt.figure(figsize=(25, 10))\ndn = dendrogram(Z)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "Z = linkage(X, 'single')\nfig = plt.figure(figsize=(25, 10))\ndn = dendrogram(Z)\nplt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-827c62b0e926b953.png" } }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-864909605f46c421.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "scipy.spatial.distance.pdist", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "scipy.spatial.distance.pdist" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "pairwise distance metrics" } ] } ], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "y", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "single" }, { "__type": "SigParam", "__tag": 4030, "name": "metric", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "euclidean" }, { "__type": "SigParam", "__tag": 4030, "name": "optimal_ordering", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "linkage" }, "references": [ ".. [1] Daniel Mullner, \"Modern hierarchical, agglomerative clustering", " algorithms\", :arXiv:`1109.2378v1`.", ".. [2] Ziv Bar-Joseph, David K. Gifford, Tommi S. Jaakkola, \"Fast optimal", " leaf ordering for hierarchical clustering\", 2001. Bioinformatics", " :doi:`10.1093/bioinformatics/17.suppl_1.S22`" ], "qa": "scipy.cluster.hierarchy._hierarchy_impl:linkage", "arbitrary": [], "local_refs": [ "Z", "method", "metric", "optimal_ordering", "y" ] }